Classical and Impulse Control for the Optimization of Dividend and Proportional Reinsurance Policies with Regime Switching

We consider the optimal proportional reinsurance and dividend strategy.
The surplus process is modeled by the classical compound Poisson risk model with
regime switching. Considering a class of utility functions, the object of the insurer
is to select the reinsurance and dividend strategy that maximizes the expected total discounted utility of the shareholders until ruin. By adapting the techniques and
methods of stochastic control, we study the quasi-variational inequality for this classical and impulse control problem and establish a verification theorem. We show that
the optimal value function is characterized as the unique viscosity solution of the
corresponding quasi-variational inequality